Buktikan limit fungsi berikut!

<p>Jawaban: Terbukti</p><p>&nbsp;</p><p>Konsep:</p><p>&gt;&gt; 1 - cos² x = sin² x</p><p>&gt;&gt; lim (x→0) (sin x)/(x) = 1</p><p>&gt;&gt; tan (-a) = -tan a</p><p>&gt;&gt; tan 60° = √3</p><p>&nbsp;</p><p>Pembahasan:</p><p>Akan dibuktikan bahwa:</p><p>lim(x → ∞) (1 - cos² 1/x)/(1/x² tan (1/x - π/3)) = -1/3 √3</p><p>Dari ruas kiri.</p><p>Misalkan 1/x = y, sehingga x = 1/y. Untuk x mendekati ∞, maka y mendekati 0.</p><p>Maka:</p><p>lim(x → ∞) (1 - cos² 1/x)/(1/x² tan (1/x - π/3))</p><p>= lim(x → ∞) (1 - cos² 1/x)/((1/x)² tan (1/x - π/3))</p><p>= lim(y → 0) (1 - cos² y)/((y)² tan (y - π/3))</p><p>= lim(y → 0) (sin² y)/(y² tan (y - π/3))</p><p>= lim(y → 0) (sin y)/(y) (sin y)/(y) (1/tan (y - π/3))</p><p>= (1) (1) (1/tan (0 - π/3))</p><p>= (1) (1) (1/tan (-π/3))</p><p>= (1) (1) (1/tan (-60°))</p><p>= (1) (1) (1/-tan 60°)</p><p>= (1) (1) (1/-√3)</p><p>= -1/√3</p><p>= -1/√3 · √3/√3</p><p>= -1/3 √3</p><p>&nbsp;</p><p>Jadi, terbukti bahwa lim(x → ∞) (1 - cos² 1/x)/(1/x² tan (1/x - π/3)) = -1/3 √3.</p>